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\begin{document}
\section{Valerie's NK Model}

\subsection{Households}

Two types, optimizing and rule-of-thumb. Utility function:
\begin{align*}
	U = E_0 \sum_{t=0}^{\infty}\beta^t \left[\ln C_t - \nu \frac{H_t^{1+\phi}}{1+\phi}\right]
\end{align*}


Constraints:
\begin{align*}
	A_t &= \frac{R_{t-1}}{\Pi_t}A_{t-1} - C_t + W_t H_t - I_t + R_t^k u_{t} K_{t-1} - T_t \\
	K_t &=  (1-\delta(u_t))K_{t-1} + I_t \left[1-S\left(\frac{I_t}{I_{t-1}}\right)\right] 
\end{align*}

FOC for optimizing households (H determined by union problem below):
\begin{align*}
	\lambda_t &= \frac{1}{C_t^o} \\
	\lambda_t &= \beta \frac{R_t}{\Pi_{t+1}} \lambda_{t+1} \\
	\lambda_t &= \mu_t \left[1-S\left(\frac{I_t^o}{I_{t-1}^o}\right)  -\frac{I_t}{I_{t-1}}S'\left(\frac{I_t^o}{I_{t-1}^o}\right)\right] + \beta \mu_{t+1}\left(\frac{I_{t+1}^o}{I_t^o}\right)^2 S'\left(\frac{I_{t+1}^o}{I_{t}^o}\right)\\
	\mu_t &= \beta\lambda_{t+1}R_{t+1}^k u_{t+1} + \beta\mu_{t+1}(1-\delta(u_{t+1})) 
%	\\
%	\lambda_t R_t^k   &= \mu_t\delta'(u_t) 
\end{align*}

Tobin's q is 
\begin{align*}
	q_t = \mu_t / \lambda_t	
\end{align*}


FOC for rule-of-thumb household:
\begin{align*}
	K_t^r=I_t^r=A_t^r&=0 \\
	C_t^r &= W_t H_t^r - T_t^r
\end{align*}





\subsection{Wages}

Union forces same hours across types:
\begin{align*}
	H_t^r = H_t^o = H_t
\end{align*}

Relative demand if reset at time $t$
\begin{align*}
	H_{t+s}^d(j) = H_{t+s}^d \left(\frac{W_{t}(j)(\frac{P_{t+s-1}}{P_{t-1}})^{\chi^w} (\frac{P_t}{P_{t+s}})}{W_{t+s}}\right)^{-\epsilon^w} = H_{t+s}^d W_{t+s}^{\epsilon^w} \left(\frac{P_{t+s}}{P_{t}}\right)^{\epsilon^w} \left(\frac{P_{t+s-1}}{P_{t-1}}\right)^{-\epsilon^w\chi^w}  W_{t}(j)^{-\epsilon^w}
\end{align*}

Optimization problem for union:
\begin{align*}
	\max_{w_t^*} \sum_{s=0}^{\infty} (\beta\theta^w)^{s}H_{t+s}^d W_{t+s}^{\epsilon^w} \left(\frac{P_{t+s}}{P_{t}}\right)^{\epsilon^w}\left(\frac{P_{t+s-1}}{P_{t-1}}\right)^{-\epsilon^w\chi^w}\left[\tilde{\lambda}_{t+s}\left(\frac{P_{t+s-1}}{P_{t-1}}\right)^{\chi^w}\left(\frac{P_{t+s}}{P_{t}}\right)^{-1}(W_t^*)^{1-\epsilon^w}   - \nu H_{t+s}^{\phi} (W_t^*)^{-\epsilon^w}\right]
\end{align*}
where $\tilde{\lambda} = (1-\gamma)\lambda_t^o + \gamma\lambda_t^r$

First order condition:
\begin{align*}
	&(\epsilon^w-1)\sum_{s=0}^{\infty} (\beta\theta^w)^{s}H_{t+s}^d W_{t+s}^{\epsilon^w} \left(\frac{P_{t+s}}{P_{t}}\right)^{\epsilon^w-1} \left(\frac{P_{t+s-1}}{P_{t-1}}\right)^{-\chi^w(\epsilon^w-1)} \tilde{\lambda}_{t+s}(W_t^*)^{1-\epsilon^w}  \\
	& = \epsilon^w \nu \sum_{s=0}^{\infty} (\beta\theta^w)^{s}H_{t+s}^d H_{t+s}^{\phi} W_{t+s}^{\epsilon^w} \left(\frac{P_{t+s}}{P_{t}}\right)^{\epsilon^w} \left(\frac{P_{t+s-1}}{P_{t-1}}\right)^{-\epsilon^w\chi^w}  (W_t^*)^{-\epsilon^w}
\end{align*}


Write recursively as:
\begin{align*}
	F_{1t} & =  \nu H_{t}^d H_{t}^{\phi} W_{t}^{\epsilon^w} (W_t^*)^{-\epsilon^w} + \nu \sum_{s=1}^{\infty} (\beta\theta^w)^{s}H_{t+s}^d H_{t+s}^{\phi} W_{t+s}^{\epsilon^w} \left(\frac{P_{t+s}}{P_{t}}\right)^{\epsilon^w} \left(\frac{P_{t+s-1}}{P_{t-1}}\right)^{-\epsilon^w\chi^w}   (W_t^*)^{-\epsilon^w} \\
	& = \nu H_{t}^d H_{t}^{\phi} W_{t}^{\epsilon^w} (W_t^*)^{-\epsilon^w} \\
	&+ \beta\theta^w \Pi_{t+1}^{\epsilon^w} \Pi_{t}^{-\chi^w\epsilon^w} \left(\frac{W_t^*}{W_{t+1}^*}\right)^{-\epsilon^w}  \nu \sum_{s=0}^{\infty} (\beta\theta^w)^{s}H_{t+1+s}^d H_{t+s}^{\phi} W_{t+1+s}^{\epsilon^w} \left(\frac{P_{t+1+s}}{P_{t+1}}\right)^{\epsilon^w} \left(\frac{P_{t+s}}{P_{t}}\right)^{-\epsilon^w\chi^w}   (W_{t+1}^*)^{-\epsilon^w} \\ 
	& = \nu H_{t}^d H_{t}^{\phi} W_{t}^{\epsilon^w} (W_t^*)^{-\epsilon^w} + \beta\theta^w \Pi_{t+1}^{\epsilon^w} \Pi_{t}^{-\chi^w\epsilon^w}  \left(\frac{W_t^*}{W_{t+1}^*}\right)^{-\epsilon^w} F_{1,t+1} \\ 
\end{align*}

\begin{align*}
	F_{2t} & =  H_{t}^d W_{t}^{\epsilon^w} \tilde{\lambda}_{t}(W_t^*)^{1-\epsilon^w} + \sum_{s=1}^{\infty} (\beta\theta^w)^{s}H_{t+s}^d W_{t+s}^{\epsilon^w} \left(\frac{P_{t+s}}{P_{t}}\right)^{\epsilon^w-1} \left(\frac{P_{t+s-1}}{P_{t-1}}\right)^{-\chi^w(\epsilon^w-1)}\tilde{\lambda}_{t+s}(W_t^*)^{1-\epsilon^w} \\ 
	& = H_{t}^d W_{t}^{\epsilon^w} \tilde{\lambda}_{t}(W_t^*)^{1-\epsilon^w} \\
	&+ \beta\theta^w  \Pi_{t+1}^{\epsilon^w-1} \Pi_{t}^{-\chi^w(\epsilon^w-1)} \left(\frac{W_t^*}{W_{t+1}^*}\right)^{1-\epsilon^w}  \sum_{s=0}^{\infty} (\beta\theta^w)^{s}H_{t+1+s}^d W_{t+1+s}^{\epsilon^w} \left(\frac{P_{t+1+s}}{P_{t+1}}\right)^{\epsilon^w-1} \left(\frac{P_{t+s}}{P_{t}}\right)^{-\chi^w(\epsilon^w-1)} \tilde{\lambda}_{t+1+s}(W_{t+1}^*)^{1-\epsilon^w} \\ 
	& = H_{t}^d W_{t}^{\epsilon^w} \tilde{\lambda}_{t}(W_t^*)^{1-\epsilon^w} + \beta\theta^w  \Pi_{t+1}^{\epsilon^w-1} \Pi_{t}^{-\chi^w(\epsilon^w-1)}  \left(\frac{W_t^*}{W_{t+1}^*}\right)^{1-\epsilon^w}  F_{2,t+1} \\ 
\end{align*}
and
\begin{align*}
	\epsilon^w  F_{1t}=(\epsilon^w-1)F_{2t}
\end{align*}

Wage distortion (labor supply higher than demand)
\begin{align*}
	H_t &= s_t^w H_t^d \\
\end{align*}

\begin{align*}
	s_t^w &= \int_0^1 \left(\frac{W_{t}(i)}{W_t}\right)^{-\epsilon^w}di \\
	&= (1-\theta^w)\left(\frac{W_t^*}{W_t}\right)^{-\epsilon^w} + \theta^w \int_0^1   \left(\frac{W_{t-1}(i)\frac{P_{t-1}}{P_t}}{W_t}\right)^{-\epsilon^w}di \\
	&=(1-\theta^w)\left(\frac{W_t^*}{W_t}\right)^{-\epsilon^w} + \theta \left(\frac{W_{t-1}}{W_t}\right)^{-\epsilon^w}\Pi_t^{\epsilon^w} s_{t-1}^w \\
\end{align*}


\subsection{Production}

Production function
\begin{align*}
	s_t Y_t = A_t (u_t K_{t-1})^{\alpha} (H_t^d)^{1-\alpha}
\end{align*}

Cost minimization
\begin{align*}
	&\min R_{t}^k u_t K_{t-1} + W_t H_t^d \\
	&\text{s.t. } A_t (u_t K_{t-1})^{\alpha} (H_t^d)^{1-\alpha} = s_t Y_t
\end{align*}

FOC:
\begin{align*}
	R_{t}^k &= \zeta_t \alpha \frac{s_t Y_t}{u_t K_{t-1}} \\
	W_t &= \zeta_t (1-\alpha) \frac{s_t Y_t}{H_t^d}
\end{align*}

Optimal capital-labor ratio:
\begin{align*}
	\frac{u_t K_{t-1}}{H_t^d} &= \frac{\alpha}{1-\alpha}\frac{W_t}{R_{t}^k}
\end{align*}

Implied total cost:
\begin{align*}
	TC_t &= R_{t}^k u_t K_{t-1} + W_t H_t^d \\
	&=R_{t}^k \left(\frac{\alpha}{1-\alpha}\frac{W_t}{R_{t}^k}\right)^{1-\alpha} \frac{s_t Y_t}{A_t} + W_t \left(\frac{\alpha}{1-\alpha}\frac{W_t}{R_{t}^k}\right)^{-\alpha}\frac{s_t Y_t}{A_t} \\
	&=\alpha^{-\alpha} (1-\alpha)^{-(1-\alpha)} (R_{t}^k)^{\alpha}W_t^{1-\alpha}\frac{s_t Y_t}{A_t} 
\end{align*}

Marginal Cost:
\begin{align*}
	MC_t &=\alpha^{-\alpha} (1-\alpha)^{-(1-\alpha)} (R_{t}^k)^{\alpha}W_t^{1-\alpha}\frac{1}{A_t} 
\end{align*}

\subsection{Prices}

Price setting problem for sticky price firm:
\begin{align*}
	\max_{p_t^*} \sum_{s=0}^{\infty} \beta^s\left(\frac{\lambda_{t+s}}{\lambda_t}\right)\theta^{s}Y_{t+s}\left[(p_t^*)^{1-\epsilon} \left(\frac{P_{t+s}}{P_t}\right)^{\epsilon-1}  - MC_{t+s} (p_t^*)^{-\epsilon}\left(\frac{P_{t+s}}{P_t}\right)^{\epsilon}\right]
\end{align*}

FOC:
%\begin{align*}
%	p_t^* &= \frac{\epsilon}{\epsilon-1}\frac{\sum_{s=0}^{\infty} \lambda_{t+s}\theta^{s} \left(\frac{P_{t+s}}{P_t}\right)^{\epsilon} MC_{t+s} }{\sum_{s=0}^{\infty} \lambda_{t+s}\theta^{s}\left(\frac{P_{t+s}}{P_t}\right)^{\epsilon-1} Y_{t+s} }
%\end{align*}
\begin{align*}
	\epsilon\sum_{s=0}^{\infty} \beta^s\left(\frac{\lambda_{t+s}}{\lambda_t}\right)\theta^{s} Y_{t+s} \left(\frac{P_{t+s}}{P_t}\right)^{\epsilon} MC_{t+s} (p_t^*)^{-\epsilon-1} 
	&=(\epsilon-1)\sum_{s=0}^{\infty} \beta^s\left(\frac{\lambda_{t+s}}{\lambda_t}\right)\theta^{s}Y_{t+s}\left(\frac{P_{t+s}}{P_t}\right)^{\epsilon-1}  (p_t^*)^{-\epsilon} 
\end{align*}

Write recursively as:
\begin{align*}
	X_{1t} & =  Y_t MC_{t} (p_t^*)^{-\epsilon-1} + \sum_{s=1}^{\infty}\beta^s \left(\frac{\lambda_{t+s}}{\lambda_t}\right)\theta^{s}Y_{t+s} \left(\frac{P_{t+s}}{P_t}\right)^{\epsilon} MC_{t+s} (p_t^*)^{-\epsilon-1} \\ 
	& = Y_{t} MC_{t} (p_t^*)^{-\epsilon-1} \\
	&+ \beta\theta \left(\frac{\lambda_{t+1}}{\lambda_t}\right)  \left(\frac{P_{t+1}}{P_{t}}\right)^{\epsilon} \left(\frac{p_t^*}{p_{t+1}^*}\right)^{-\epsilon-1}  \sum_{s=0}^{\infty}\beta^s \left(\frac{\lambda_{t+1+s}}{\lambda_{t+1}}\right)\theta^{s} Y_{t+s+1} \left(\frac{P_{t+1+s}}{P_{t+1}}\right)^{\epsilon} MC_{t+1+s} (p_{t+1}^*)^{-\epsilon-1} \\ 
	& = Y_{t} MC_{t} (p_t^*)^{-\epsilon-1} + \beta\theta  \left(\frac{\lambda_{t+1}}{\lambda_{t}}\right)\left(\frac{P_{t+1}}{P_{t}}\right)^{\epsilon} \left(\frac{p_t^*}{p_{t+1}^*}\right)^{-\epsilon-1}  X_{1,t+1} \\ 
\end{align*}

\begin{align*}
	X_{2t} & =  Y_t  (p_t^*)^{-\epsilon} + \sum_{s=1}^{\infty} \beta^s\left(\frac{\lambda_{t+s}}{\lambda_t}\right)\theta^{s}Y_{t+s}\left(\frac{P_{t+s}}{P_t}\right)^{\epsilon-1}  (p_t^*)^{-\epsilon}  \\ 
	& =  Y_{t} (p_t^*)^{-\epsilon} \\
	& + \beta\theta \left(\frac{\lambda_{t+1}}{\lambda_t}\right)  \left(\frac{P_{t+1}}{P_{t}}\right)^{\epsilon-1} \left(\frac{p_t^*}{p_{t+1}^*}\right)^{-\epsilon}  \sum_{s=0}^{\infty} \beta^s\left(\frac{\lambda_{t+1+s}}{\lambda_{t+1}}\right)\theta^{s} Y_{t+s+1} \left(\frac{P_{t+1+s}}{P_{t+1}}\right)^{\epsilon-1} (p_{t+1}^*)^{-\epsilon} \\ 
	& =  Y_{t} (p_t^*)^{-\epsilon} + \beta\theta  \left(\frac{\lambda_{t+1}}{\lambda_{t}}\right)\left(\frac{P_{t+1}}{P_{t}}\right)^{\epsilon-1} \left(\frac{p_t^*}{p_{t+1}^*}\right)^{-\epsilon}  X_{2,t+1} \\ 
\end{align*}
and
\begin{align*}
	\epsilon  X_{1t}=(\epsilon-1)X_{2t}
\end{align*}


Aggregate Prices and inflation:
\begin{align*}
	P_t^{1-\epsilon} &= (1-\theta)(P_{t}^*)^{1-\epsilon} + \theta P_{t-1}^{1-\epsilon} \\
	1 &= (1-\theta)(p_{t}^*)^{1-\epsilon} + \theta \Pi_{t}^{-(1-\epsilon)} \\
\end{align*}

Output distortion
\begin{align*}
	s_t &= \int_0^1 \left(\frac{P_{t}(i)}{P_t}\right)^{-\epsilon}di \\
	&= (1-\theta)(p_t^*)^{-\epsilon} + \theta \int_0^1   \left(\frac{P_{t-1}(i)}{P_t}\right)^{-\epsilon}di \\
	&=(1-\theta)(p_t^*)^{-\epsilon} + \theta \Pi_t^{\epsilon} s_{t-1} \\
\end{align*}


\subsection{Government}

Interest rate rule
\begin{align*}
	R_t &= (1-\rho_r)R_{t-1} + \rho_r\left[R + \phi_\pi(\Pi_t-\bar{\Pi}) + \phi_y\left(\frac{Y_t}{\bar{Y}}-1\right)\right]
\end{align*}

Budget constraint
\begin{align*}
	B_t &= \frac{R_{t-1}}{\Pi_t}B_{t-1} + G_t - T_t
\end{align*}


Budget rule:
\begin{align*}
	T_t &= T + \phi_b(B_{t-k}-\bar{B}) - \epsilon_t
\end{align*}

Assume same transfers:
\begin{align*}
	T_t^r = T_t^o
\end{align*}

\subsection{Market Clearing}

Aggregates:
\begin{align*}
	C_t &= (1-\gamma) C_t^o + \gamma C_t^r \\
	H_t &= (1-\gamma) H_t^o + \gamma H_t^r \\
	I_t &= (1-\gamma) I_t^o + \gamma I_t^r \\
	K_t &= (1-\gamma) K_t^o + \gamma K_t^r \\
	A_t &= (1-\gamma) A_t^o + \gamma A_t^r \\
	T_t &= (1-\gamma) T_t^o + \gamma T_t^r \\
\end{align*}

Goods market
\begin{align*}
	Y_t &= C_t + I_t + G_t
\end{align*}


\subsection{Functional Forms}

\begin{align*}
	\delta(u_t) &= \delta_0 + \delta_1 (u_t-1) + \delta_2 (u_t-1)^2 \\
	S\left(\frac{I_t}{I_{t-1}}\right) &=\frac{\kappa}{2}\left(\frac{I_t}{I_{t-1}}-1\right)^{2}
\end{align*}


\end{document}